Problem: Let $\alpha$ and $\beta$ be the roots of $x^2 + px + 1 = 0,$ and let $\gamma$ and $\delta$ are the roots of $x^2 + qx + 1 = 0.$  Express
\[(\alpha - \gamma)(\beta - \gamma)(\alpha + \delta)(\beta + \delta)\]in terms of $p$ and $q.$
Explanation: Since $\alpha$ and $\beta$ are the roots of $x^2 + px + 1 = 0,$
\[(x - \alpha)(x - \beta) = x^2 + px + 1.\]Setting $x = \gamma,$ we get
\[(\gamma - \alpha)(\gamma - \beta) = \gamma^2 + p \gamma + 1.\]or $(\alpha - \gamma)(\beta - \gamma) = \gamma^2 + p \gamma + 1.$

Setting $x = -\delta,$ we get
\[(-\delta - \alpha)(-\delta - \beta) = \delta^2 - p \delta + 1,\]or $(\alpha + \beta)(\beta + \delta) = \delta^2 - p \delta + 1.$

Since $\gamma$ and $\delta$ are the roots of $x^2 + qx + 1 = 0,$ $\gamma^2 + q \gamma + 1 = 0$ and $\delta^2 + q \delta + 1 = 0.$  Then
\[\gamma^2 + p \gamma + 1 = (p - q) \gamma\]and
\[\delta^2 - p \delta + 1 = -(p + q) \delta.\]Finally, by Vieta's formulas, $\gamma \delta = 1,$ so
\[(p - q) \gamma \cdot (-(p + q)) \delta = (q - p)(q + p) = \boxed{q^2 - p^2}.\]